Math 182 Name: __ Key __

Exam 1 February 20, 2013

**Show all work-
Answers that are not fully justified will earn little or no credit**

**Be sure to include units in your answers
where appropriate.**

**If a problem asks for an exact
answer, then a decimal approximation is not sufficient (so don't round
off **

1.
Find the *exact*
value of each of the following limits.
Note: You may use numerical and/or graphical means to check your
answers, but these techniques will not be sufficient to justify that your
answers are exact.

*This is continuous at the point, so just
evaluate: *

*This limit could also be found by algebraic
means.*

*Letting x=0, we see
A=3*

*Letting x=-1, we see
-1C=2, or C=-2*

*Letting x=1, we see
4A+2B+C=4, so 12+2B-2=4, so B=-3*

3. The graph shown is y=f(t). .

Find the following values. You may approximate if

necessary, but must justify your answers.

a.

*The area under the curve
from 0 to 2 is about 10*

*(note
that the shape is very nearly a trapezoid) *

b.

c. The x-values of g’s local maxima, if any.

g’ local extrema occur at x=6 and x=7.

At x=6 g’ changes from positive to negative, so g has a local maximum there.

4. a. Evaluate

b. A particle moves along a straight line
with a velocity of v(t) = (t×sin t) during the time 0 Ł t Ł 2p , where t is measured in seconds, and v(t) in feet/second. Positive velocity corresponds to moving to
the right, and negative velocity to the left.
At the end of the 2p seconds, where is the particle relative to where it
started?

, so the particle is 2π feet to the left of its starting point.

5. a. Evaluate

b. A tree sapling is initially 2 feet tall, and grows at a rate of feet/year.

Carefully INTERPRET in one or two complete sentences the significance of the answer to part (a) in this context. Full credit can be earned for this interpretation even if you did not get the correct answer to part (a).

*The definite integral
of a rate of change of a function over some interval gives the total change of
the original function over that interval.
In this case, the integral would represent the total amount of growth of
this tree from the time it is planted until ∞ (forever). *

*So the tree will grow
another 5 feet, to an ultimate height of 8 feet if it lives forever. *

6. Determine whether the following integrals converge or diverge. For any that converge, find the value:

*So it converges to 2.*

*:
handling the first of these integrals first (although we could start
with either), **, so the integral diverges.*

7. The function shown is

a. Draw on this graph the 2
rectangles used to approximate the integral of this function from -1 to 3 using
the M_{2} method. Then calculate
(by hand) the value of the M_{2} approximation.

Value of M_{2 }:__ 2*1+2*4=10 .__

b.
Use your calculator to find the M_{10}
and T_{10} approximations to this integral (round to the nearest
thousandth).

Value of M_{10 }:__ 10.786 .__

Value of T_{10 }:__ 10.889 .__

c.
Recall the error bound for the trapezoid
approximation: . __Use this formula__ with an appropriately
chosen value for K to determine the maximum possible error in the T_{10}
approximation above.

*(Hint:
the derivative of a function of the form **). *

So K can be chosen to be exactly 8ln(2)^{2} or some convenient number suitably close *and larger than this* –e.g. 4 would be a
good choice.

Extra Credit: Does the integral converge or diverge? Carefully justify your answer.

*Answer 1 (easiest): **, so the comparison
test applies, and tells us that the desired integral must converge too, to a
number between 0 and 1.*

*Answer 2 (longer):
Substitute **so the integral becomes
*

. The first of these
terms is 0 (use L’hopitals rule, or simply notice
that as b gets larger and larger, you are taking the ln
of a number closer and closer to 1, so the ln would
approach 0), so the final answer must be –ln(1/2) = ln(2).